Exercises

1. Factor fully.
   1. \(x^2-9\)
   2. \(16x^2-25\)
   3. \(81-x^2\)
   4. \(12x^3-3xy^2\)
   5. \(5x^8-405\)
2. 1. Using integer coefficients, determine two possible standard form quadratic relations that have zeros \(x=\pm \dfrac{7}{3}\). 
   2. Determine all possible standard form quadratic relations that have zeros \(x=\pm \dfrac{q}{p}\) for all real numbers \(\)\(p\) and \(q\) with \(p \ne 0\).
3. Factor fully.
   1. \(x^2-8x+16\)
   2. \(16x^2+24x+9\)
   3. \(2x^3+40x^2+200x\)
   4. \(3x^3y-24x^2y+48xy\)
4. Determine the missing terms, given that each of the following expressions is a perfect square. Write the factored form of each expression.
   1. \(\boxed{\phantom{\square\square}}-12xy+9y^2\)
   2. \(49x^6+42x^3y^5 +\boxed{\phantom{\square\square}}\)
   3. \(100x^6-300x^3+\boxed{\phantom{\square\square}}\)
5. Complete the table.

   Relation	Factored Form	Zero(s)	Axis of Symmetry	Vertex
   \(y=x^2-9\)
   \(y=x^2-12x+36\)
   \(y=9x^2+12x+4\)

6. Factor fully by grouping.
   1. \(10x^2+2xy-15x-3y\)
   2. \(3x^2+2xy-9x-6y\)
   3. \(15xy+1+3x+5y\)
7. Use substitution to factor fully.
   1. \((x^2+2x)^2-2(x^2+2x)-3\)
   2. \((y^4+y^2)^2-92(y^4+y^2)+180\)
8. The relation \(y=(x^2+2x)^2-2(x^2+2x)-3\) is not quadratic, yet it is still possible to determine the zeros. Determine the zeros of \(y=(x^2+2x)^2-2(x^2+2x)-3\).
9. 1. Using integer coefficients, determine two possible standard form quadratic relations that have exactly one zero, namely \(x=-\dfrac{3}{4}\).
   2. Determine all possible standard form quadratic relations that have exactly one zero, namely \(x=\dfrac{q}{p}\), for all real numbers \(\)\(p\) and \(q\) with \(p \ne 0\).
10. One factor of \(x^3+3x^2-4\) is \(x-1\). Factor \(x^3+3x^2-4\) fully.